On the thermomechanical consistency of the time differential dual-phase-lag models of heat conduction

TL;DR

This paper examines the thermodynamic consistency of time differential dual-phase-lag heat conduction models, highlighting restrictions related to differential operator properties and proposing reformulation within fading memory theory to ensure physical plausibility.

Contribution

It identifies conditions under which dual-phase-lag models are thermodynamically consistent and reformulates the models using fading memory theory for improved physical validity.

Findings

Models with approximation order ≥5 exhibit instabilities due to complex roots with positive real parts.

For orders ≤4, models are compatible with the Second Law of Thermodynamics.

Reformulation within fading memory theory ensures thermodynamic consistency.

Abstract

This paper deals with the time differential dual-phase-lag heat transfer models aiming, at first, to identify the eventually restrictions that make them thermodynamically consistent. At a first glance it can be observed that the capability of a time differential dual-phase-lag model of heat conduction to describe real phenomena depends on the properties of the differential operators involved in the related constitutive equation. In fact, the constitutive equation is viewed as an ordinary differential equation in terms of the heat flux components (or in terms of the temperature gradient) and it results that, for approximation orders greater than or equal to five, the corresponding characteristic equation has at least a complex root having a positive real part. That leads to a heat flux component (or temperature gradient) that grows to infinity when the time tends to infinity and so there occur some instabilities. Instead, when the approximation orders are lower than or equal to four, this is not the case and there is the need to study the compatibility with the Second Law of Thermodynamics. To this aim the related constitutive equation is reformulated within the system of the fading memory theory, and thus the heat flux vector is written in terms of the history of the temperature gradient and on this basis the compatibility of the model with the thermodynamical principles is analyzed.